Question

Linear Programming:
A real estate developer is planning a new mini apartment complex.

Three types of units can be built: one-bedroom apartments, two-bedroom apartments, and three-bedroom apartments.

Each one-bedroom apartment requires 700 square feet; each two-bedroom apartment requires 850 square feet; and each three-bedroom apartment requires 1,250 square feet.
The developer wants to keep a mix of apartment types in the complex. He believes that the number of one-bedroom apartments should be at least 15% of the total number of apartments. However three-bedroom apartments should not be more than 30% of the total number of apartments.

Local zoning laws do not allow the developer to build more than 42 units in this particular building location, and restrict the building to a maximum of 36,000 square feet.

Market studies show that one-bedrooms rent for $825 per month, two-bedrooms for $1,225 per month, and three-bedrooms for $1,775 per month.


a)Determine the constraint for square feet.

b)Determine the constraint for the number of total units.

c) Determine the constraint for one-bedroom apartments.​​​​​​​

d) Determine the objective function.

Answer 1

Let's assume the number of one-bedroom apartments is X, the number of two-bedroom apartments is Y and the number of three-bedroom apartments is Z.

a) Total square feet available is 36,000 sq ft

Total square feet required for the apartments = 700 X + 850 Y + 1250 Z

So, the constraint for square feet is as follows:

700 X + 850 Y + 1250 Z < = 36,000

b) Total number of units allowed = 42

So, the constraint for the number of total units is as follows:

X + Y + Z <= 42

c) The number for one-bedroom apartments should be at least 15% of the total number of apartments.

So,

X >= 15 % of ( X + Y + Z)

X >= 15 ( X + Y + Z) / 100

100 X >= 15 X + 15 Y + 15 Z

85 X -15 Y -15 Z >= 0 , or 15 Y + 15 Z - 85 X <= 0

Therefore, the contraint for one-bedroom paprtments is as follows:

15 Y + 15 Z - 85 X <= 0

d) The objective function should be the maximization of rent. Let's assume the objective function is G

Total rent = 825 X + 1225 Y + 1775 Z

Therefore, the objective function is as follows:

Max G = 825 X + 1225 Y + 1775 Z